Dimension free $L^p$-bounds of maximal functions associated to products of Euclidean balls

TL;DR

This paper extends Bourgain's dimension-free $L^p$ bounds for maximal functions from cubes to products of Euclidean balls and spheres, providing new bounds for various geometric configurations.

Contribution

It generalizes dimension-free $L^p$ bounds to products of Euclidean balls and spheres, broadening the scope of Bourgain's and Stein's results.

Findings

Dimension-free $L^p$ bounds for products of Euclidean balls.

Dimension-free $L^p$ bounds for products of Euclidean spheres for $p > N/(N-1)$.

Application of rotation methods to derive bounds for spherical maximal functions.

Abstract

A few years ago, Bourgain proved that the centered Hardy-Littlewood maximal function for the cube has dimension free $L^p$-bounds for $p>1$. We extend his result to products of Euclidean balls of different dimensions. In addition, we provide dimension free $L^p$-bounds for the maximal function associated to products of Euclidean spheres for $p > \frac{N}{N-1}$ and $N \ge 3$, where $N-1$ is the lowest occurring dimension of a single sphere. The aforementioned result is obtained from the latter one by applying the method of rotations from Stein's pioneering work on the spherical maximal function.